Minimum flows in the total graph of a finite commutative ring

Abstract

Let $R$ be a commutative ring with zero-divisor set $Z(R)$. The total graph of $R$, denoted by $T(Gamma(R))$, is the simple (undirected) graph with vertex set $R$ where two distinct vertices are adjacent if their sum lies in $Z(R)$. This work considers minimum zero-sum $k$-flows for $T(Gamma(R))$. Both for $vert Rvert$ even and the case when $vert Rvert$ is odd and $Z(G)$ is an ideal of $R$ it is shown that $T(Gamma(R))$ has a zero-sum $3$-flow, but no zero-sum $2$-flow. As a step towards resolving the remaining case, the total graph $T(Gamma(mathbb{Z}_n ))$ for the ring of integers modulo $n$ is considered. Here, minimum zero-sum $k$-flows are obtained for $n = p^r$ and $n = p^r q^s$ (where $p$ and $q$ are primes, $r$ and $s$ are positive integers). Minimum zero-sum $k$-flows as well as minimum constant-sum $k$-flows in regular graphs are also investigated.

Keywords

• Constant-sum k-flow
• minimum flow
• the ring of integers modulo n
• total graph of a commutative ring
• zero-sum k-flow